Forces in vector form: equilibrium (find unknowns)

3. Three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) acting on a particle \(P\) are given by $$\begin{aligned} & \mathbf { F } _ { 1 } = ( 7 \mathbf { i } - 9 \mathbf { j } ) \mathrm { N } \\ & \mathbf { F } _ { 2 } = ( 5 \mathbf { i } + 6 \mathbf { j } ) \mathrm { N } \\ & \mathbf { F } _ { 3 } = ( p \mathbf { i } + q \mathbf { j } ) \mathrm { N } \end{aligned}$$ where \(p\) and \(q\) are constants.
Given that \(P\) is in equilibrium,

1. find the value of \(p\) and the value of \(q\). The force \(\mathbf { F } _ { 3 }\) is now removed. The resultant of \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) is \(\mathbf { R }\). Find
2. the magnitude of \(\mathbf { R }\),
3. the angle, to the nearest degree, that the direction of \(\mathbf { R }\) makes with \(\mathbf { j }\).

1. Three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) act on a particle \(P\).

$$\mathbf { F } _ { 1 } = ( 2 \mathbf { i } + 3 a \mathbf { j } ) \mathrm { N } ; \quad \mathbf { F } _ { 2 } = ( 2 a \mathbf { i } + b \mathbf { j } ) \mathrm { N } ; \quad \mathbf { F } _ { 3 } = ( b \mathbf { i } + 4 \mathbf { j } ) \mathrm { N } .$$ The particle \(P\) is in equilibrium under the action of these forces.
Find the value of \(a\) and the value of \(b\).

3. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors.] Three forces, \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\), are given by $$\mathbf { F } _ { 1 } = ( 5 \mathbf { i } + 2 \mathbf { j } ) \mathrm { N } \quad \mathbf { F } _ { 2 } = ( - 3 \mathbf { i } + \mathbf { j } ) \mathrm { N } \quad \mathbf { F } _ { 3 } = ( a \mathbf { i } + b \mathbf { j } ) \mathrm { N }$$ where \(a\) and \(b\) are constants.
The forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) act on a particle \(P\) of mass 4 kg .
Given that \(P\) rests in equilibrium on a smooth horizontal surface under the action of these three forces,

1. find the size of the angle between the direction of \(\mathbf { F } _ { 3 }\) and the direction of \(- \mathbf { j }\). The force \(\mathbf { F } _ { 3 }\) is now removed and replaced by the force \(\mathbf { F } _ { 4 }\) given by \(\mathbf { F } _ { 4 } = \lambda ( \mathbf { i } + 3 \mathbf { j } )\) N, where \(\lambda\) is a positive constant. When the three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 4 }\) act on \(P\), the acceleration of \(P\) has magnitude \(3.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
2. Find the value of \(\lambda\).
   

3 A particle rests on a smooth, horizontal plane. Horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) lie in this plane. The particle is in equilibrium under the action of the three forces \(( - 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { N }\) and \(( 21 \mathbf { i } - 7 \mathbf { j } ) \mathrm { N }\) and \(\mathbf { R N }\).

1. Write down an expression for \(\mathbf { R }\) in terms of \(\mathbf { i }\) and \(\mathbf { j }\).
2. Find the magnitude of \(\mathbf { R }\) and the angle between \(\mathbf { R }\) and the \(\mathbf { i }\) direction.

3 A particle is in equilibrium under the action of three forces in newtons given by $$\mathbf { F } _ { 1 } = \binom { 8 } { 0 } , \quad \mathbf { F } _ { 2 } = \binom { 2 a } { - 3 a } \quad \text { and } \quad \mathbf { F } _ { 3 } = \binom { 0 } { b } .$$ Find the values of the constants \(a\) and \(b\).

4 In this question, the \(x\) and \(y\) directions are horizontal and vertically upwards respectively.
A particle of mass 1.5 kg is in equilibrium under the action of its weight and forces \(\mathbf { F } _ { 1 } = \binom { 4 } { - 2 } \mathrm {~N}\) and \(\mathbf { F } _ { 2 }\). and \(\mathbf { F } _ { 2 }\).

1. Find the force \(\mathbf { F } _ { 2 }\). The force \(\mathbf { F } _ { 2 }\) is changed to \(\binom { 2 } { 20 } \mathrm {~N}\).
2. Find the acceleration of the particle.

3 Forces \(\mathbf { F } _ { 1 } = ( 2 \mathbf { i } + 9 \mathbf { j } ) \mathbf { N }\) and \(\mathbf { F } _ { 2 } = ( - \mathbf { i } + \mathbf { j } ) \mathbf { N }\) act on a particle. A third force \(\mathbf { F } _ { 3 }\) acts so that the particle is in equilibrium under the action of the three forces. Find the force \(\mathbf { F } _ { 3 }\).

1. Three forces \(( - 5 \mathbf { i } + 4 p \mathbf { j } ) \mathrm { N } , ( 2 q \mathbf { i } + 3 \mathbf { j } ) \mathrm { N }\) and \(( \mathbf { i } + \mathbf { j } ) \mathrm { N }\) act on a particle \(A\) of mass 2 kg .

Given that \(A\) is in equilibrium, find the values of \(p\) and \(q\).

1. The resultant of two forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) is \(( - 2 \mathbf { i } + 9 \mathbf { j } ) \mathrm { N }\).

Given that \(\mathbf { F } _ { \mathbf { 1 } } = ( 2 p \mathbf { i } - 3 q \mathbf { j } ) \mathrm { N }\) and \(\mathbf { F } _ { \mathbf { 2 } } = ( 5 q \mathbf { i } + 4 p \mathbf { j } ) \mathrm { N }\), calculate the values of \(p\) and \(q\).
(5 marks)

5 A cylindrical tub of mass 250 kg is on a horizontal floor. Resistance to its motion other than that due to friction is negligible. The first attempt to move the tub is by pulling it with a force of 150 N in the \(\mathbf { i }\) direction, as shown in Fig. 8.1. \begin{figure}[h] \end{figure}

1. Calculate the acceleration of the tub if friction is ignored. In fact, there is friction and the tub does not move.
2. Write down the magnitude and direction of the frictional force opposing the pull. Two more forces are now added to the 150 N force in a second attempt to move the tub, as shown in Fig. 8.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5a1895e1-abe3-4739-876a-f19458f0f6ed-4_497_927_1350_646} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
   \end{figure} Angle \(\theta\) is acute and chosen so that the resultant of the three forces is in the \(\mathbf { i }\) direction.
3. Determine the value of \(\theta\) and the resultant of the three forces. With this resultant force, the tub moves with constant acceleration and travels 1 metre from rest in 2 seconds.
4. Show that the magnitude of the friction acting on the tub is 661 N , correct to 3 significant figures. When the speed of the tub is \(1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it comes to a part of the floor where the friction on the tub is 200 N greater. The pulling forces stay the same.
5. Find the velocity of the tub when it has moved a further 1.65 m .

1 A particle rests on a smooth, horizontal plane. Horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) lie in this plane. The particle is in equilibrium under the action of the three forces \(( - 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { N }\) and \(( 21 \mathbf { i } - 7 \mathbf { j } ) \mathrm { N }\) and \(\mathbf { R N }\).

1. Write down an expression for \(\mathbf { R }\) in terms of \(\mathbf { i }\) and \(\mathbf { j }\).
2. Find the magnitude of \(\mathbf { R }\) and the angle between \(\mathbf { R }\) and the \(\mathbf { i }\) direction.

Force \(\mathbf{F}\) is \(\begin{pmatrix} 4 \\ 1 \\ 2 \end{pmatrix}\) N and force \(\mathbf{G}\) is \(\begin{pmatrix} -6 \\ 2 \\ 4 \end{pmatrix}\) N.

1. Find the resultant of \(\mathbf{F}\) and \(\mathbf{G}\) and calculate its magnitude. [4]
2. Forces \(\mathbf{F}\), \(2\mathbf{G}\) and \(\mathbf{H}\) act on a particle which is in equilibrium. Find \(\mathbf{H}\). [3]

A particle, under the action of two constant forces, is moving across a perfectly smooth horizontal surface at a constant speed of \(10 \text{ m s}^{-1}\) The first force acting on the particle is \((400\mathbf{i} + 180\mathbf{j})\) N. The second force acting on the particle is \((p\mathbf{i} - 180\mathbf{j})\) N. Find the value of \(p\). Circle your answer. [1 mark] \(-400\) \quad \(-390\) \quad \(390\) \quad \(400\)

Two forces, \(\mathbf{F_1}\) and \(\mathbf{F_2}\), are acting on a particle of mass 3 kilograms. It is given that $$\mathbf{F_1} = \begin{pmatrix} a \\ 23 \end{pmatrix} \text{ newtons and } \mathbf{F_2} = \begin{pmatrix} 4 \\ b \end{pmatrix} \text{ newtons}$$ where \(a\) and \(b\) are constants. The particle has an acceleration of \(\begin{pmatrix} 4b \\ a \end{pmatrix}\) m s\(^{-2}\) Find the value of \(a\) and the value of \(b\) [4 marks]

A particle \(P\) of mass 0.5 kg is in equilibrium under the action of three forces \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\). $$\mathbf{F}_1 = (9\mathbf{i} + 6\mathbf{j} - 12\mathbf{k})\text{N} \quad \text{and} \quad \mathbf{F}_2 = (6\mathbf{i} - 7\mathbf{j} + 3\mathbf{k})\text{N}.$$

1. Find the force \(\mathbf{F}_3\). [2]
2. Forces \(\mathbf{F}_2\) and \(\mathbf{F}_3\) are removed so that \(P\) moves in a straight line \(AB\) under the action of the single force \(\mathbf{F}_1\). The points \(A\) and \(B\) have position vectors \((2\mathbf{i} - 9\mathbf{j} + 7\mathbf{k})\) m and \((8\mathbf{i} - 5\mathbf{j} - \mathbf{k})\) m respectively. The particle \(P\) is initially at rest at \(A\).
   1. Verify that \(\mathbf{F}_1\) acts parallel to the vector \(\overrightarrow{AB}\).
   2. Find the work done by the force \(\mathbf{F}_1\) as the particle moves from \(A\) to \(B\).
   3. By using the work-energy principle, find the speed of \(P\) as it reaches \(B\). [7]

Three forces \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) acting on a particle are given by $$\mathbf{F}_1 = (3\mathbf{i} - 2a\mathbf{j})\text{N}, \quad \mathbf{F}_2 = (2b\mathbf{i} + 3a\mathbf{j})\text{N} \quad \text{and} \quad \mathbf{F}_3 = (-2\mathbf{i} + b\mathbf{j})\text{N}.$$ The particle is in equilibrium under the action of these three forces. Find the value of \(a\) and the value of \(b\). [3]

A particle is being held in equilibrium by the following set of forces (in newtons). $$\mathbf{F}_1 = 5\mathbf{i} - 8\mathbf{j}, \quad \mathbf{F}_2 = -3\mathbf{i} - 4\mathbf{j}, \quad \mathbf{F}_3 = 6\mathbf{i} + 6\mathbf{j} \quad \text{and} \quad \mathbf{F}_4.$$

1. Find \(\mathbf{F}_4\) in terms of \(\mathbf{i}\) and \(\mathbf{j}\). [2]
2. Hence find the magnitude and direction of \(\mathbf{F}_4\). [4]

A particle \(P\) of mass \(1.5\) kg is placed on a smooth horizontal table. The particle is initially at the origin of a \(2\)-dimensional vector system defined by perpendicular unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) in the plane of the table. The particle is subject to three forces of magnitudes \(10\) N, \(12\) N and \(F\) N, acting in the directions of the vectors \(3\mathbf{i} + 4\mathbf{j}\), \(-\mathbf{j}\) and \(-\cos \theta \mathbf{i} + \sin \theta \mathbf{j}\) respectively, and no others.

1. Given that the system is in equilibrium, determine \(F\) and \(\theta\). [6]

The force of magnitude \(12\) N is replaced by one of magnitude \(4\) N, but in the opposite direction. The particle is initially at rest.

1. Find the position vector of the particle \(3\) seconds later. [5]